
-
Previous Article
Wave-propagation in an incompressible hollow elastic cylinder with residual stress
- DCDS-S Home
- This Issue
-
Next Article
Dirichlet problem for a diffusive logistic population model with two delays
Prey-predator model with nonlocal and global consumption in the prey dynamics
1. | Department of Mathematics & Statistics, Indian Institute of Technology Kanpur, Kanpur - 208016, India |
2. | Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, 69622 Villeurbanne, France |
3. | INRIA, Université de Lyon, Université Lyon 1, Institut Camille Jordan, 43 Bd. du 11 Novembre 1918, 69200 Villeurbanne Cedex, France |
4. | Peoples Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation |
A prey-predator model with a nonlocal or global consumption of resources by prey is studied. Linear stability analysis about the homogeneous in space stationary solution is carried out to determine the conditions of the bifurcation of stationary and moving pulses in the case of global consumption. Their existence is confirmed in numerical simulations. Periodic travelling waves and multiple pulses are observed for the nonlocal consumption.
References:
[1] |
A. Apreutesei, A. Ducrot and V. Volpert,
Competition of species with intra-specific competition, Math. Model. Nat. Phenom., 3 (2008), 1-27.
doi: 10.1051/mmnp:2008068. |
[2] |
N. Apreutesei, A. Ducrot and V. Volpert,
Travelling waves for integro-differential equations in population dynamics, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), 541-561.
doi: 10.3934/dcdsb.2009.11.541. |
[3] |
N. Apreutesei, N. Bessonov, V. Volpert and V. Vougalter,
Sptaial structures and generalized travelling waves for an integro-differential equation, Discrete Contin. Dyn. Syst. Ser. B, 13 (2010), 537-557.
doi: 10.3934/dcdsb.2010.13.537. |
[4] |
O. Aydogmus,
Patterns and transitions to instability in an intraspecific competition model with nonlocal diffusion and interaction, Math. Model. Nat. Phenom., 10 (2015), 17-19.
doi: 10.1051/mmnp/201510603. |
[5] |
M. Banerjee and S. Banerjee,
Turing instabilities and spatio-temporal chaos in ratio-dependent Holling-Tanner model, Math. Biosci., 236 (2012), 64-76.
doi: 10.1016/j.mbs.2011.12.005. |
[6] |
M. Banerjee, N. Mukherjee and V. Volpert,
Prey-predator model with a nonlocal bistable dynamics of prey, Mathematics, 6 (2018), 1-13.
doi: 10.3390/math6030041. |
[7] |
M. Banerjee and S. Petrovskii, Self-organized spatial patterns and chaos in a ratio-dependent predator-prey system, Theor. Ecol., 4 (2011), 37-53. Google Scholar |
[8] |
M. Banerjee and V. Volpert, Prey-predator model with a nonlocal consumption of prey, Chaos, 26 (2016), 12pp.
doi: 10.1063/1.4961248. |
[9] |
M. Banerjee and V. Volpert, Spatio-temporal pattern formation in Rosenzweig-McArthur model: Effect of nonlocal interactions, Ecol. Complex., 30 (2017), 2-10. Google Scholar |
[10] |
M. Banerjee, V. Vougalter and V. Volpert,
Doubly nonlocal reaction–diffusion equations and the emergence of species, Appl. Math. Model., 42 (2017), 591-599.
doi: 10.1016/j.apm.2016.10.041. |
[11] |
M. Baurmann, W. Ebenhoh and U. Feudel,
Turing instabilities and pattern formation in a benthic nutrient-microorganism system, Math. Biosci. Eng., 1 (2004), 111-130.
doi: 10.3934/mbe.2004.1.111. |
[12] |
M. Baurmann, T. Gross and U. Feudel,
Instabilities in spatially extended predator-prey systems: Spatio-temporal patterns in the neighborhood of Turing-Hopf bifurcations, J. Theoret. Biol., 245 (2007), 220-229.
doi: 10.1016/j.jtbi.2006.09.036. |
[13] |
A. Bayliss and V. A. Volpert,
Patterns for competing populations with species specific nonlocal coupling, Math. Model. Nat. Phenom., 10 (2015), 30-47.
doi: 10.1051/mmnp/201510604. |
[14] |
N. Bessonov, N. Reinberg and V. Volpert,
Mathematics of Darwin's diagram, Math. Model. Nat. Phenom., 9 (2014), 5-25.
doi: 10.1051/mmnp/20149302. |
[15] |
S. Fasani and S. Rinaldi,
Factors promoting or inhibiting Turing instability in spatially extended prey-predator systems, Ecol. Model., 222 (2011), 3449-3452.
doi: 10.1016/j.ecolmodel.2011.07.002. |
[16] |
T. Galochkina, M. Marion and V. Volpert,
Initiation of reaction-diffusion waves of blood coagulation, Phys. D, 376-377 (2018), 160-170.
doi: 10.1016/j.physd.2017.11.006. |
[17] |
G. F. Gause, The Struggle for Existence, Williams and Wilkins, Baltimore, 1934.
doi: 10.5962/bhl.title.4489. |
[18] |
S. Genieys, N. Bessonov and V. Volpert,
Mathematical model of evolutionary branching, Math. Comput. Modelling, 49 (2009), 2109-2115.
doi: 10.1016/j.mcm.2008.07.018. |
[19] |
S. Genieys, V. Volpert and P. Auger,
Pattern and waves for a model in population dynamics with nonlocal consumption of resources, Math. Model. Nat. Phenom., 1 (2006), 65-82.
doi: 10.1051/mmnp:2006004. |
[20] |
S. Genieys, V. Volpert and P. Auger,
Adaptive dynamics: Modelling Darwin's divergence principle, Comp. Ren. Biol., 329 (2006), 876-879.
doi: 10.1016/j.crvi.2006.08.006. |
[21] |
J. D. Murray, Mathematical Biology. Ⅱ: Spatial Models And Biomedical Applications, Interdisciplinary Applied Mathematics, 19, Springer-Verlag, New York, 2003.
doi: 10.1007/b98869. |
[22] |
S. Pal, S. Ghorai and M. Banerjee,
Analysis of a prey-predator model with non-local interaction in the prey population, Bull. Math. Biol., 80 (2018), 906-925.
doi: 10.1007/s11538-018-0410-x. |
[23] |
S. V. Petrovskii and H. Malchow,
A minimal model of pattern formation in a prey-predator system, Math. Comput. Modelling, 29 (1999), 49-63.
doi: 10.1016/S0895-7177(99)00070-9. |
[24] |
J. A. Sherratt, B. T. Eagan and M. A. Lewis,
Oscillations and chaos behind predator-prey invasion: Mathematical artifact or ecological reality?, Phil. Trans. R. Soc. Lond. B, 352 (1997), 21-38.
doi: 10.1098/rstb.1997.0003. |
[25] |
V. Volpert, Branching and aggregation in self-reproducing systems, in MMCS, Mathematical Modelling of Complex Systems, ESAIM Proc. Surveys, 47, EDP Sci., Les Ulis, 2014,116–129.
doi: 10.1051/proc/201447007. |
[26] |
V. Volpert, Elliptic Partial Differential Equations, Monographs in Mathematics, 104, Birkhäuser/Springer Basel AG, Basel, 2014.
doi: 10.1007/978-3-0348-0813-2. |
[27] |
V. Volpert,
Pulses and waves for a bistable nonlocal reaction-diffusion equation, Appl. Math. Lett., 44 (2015), 21-25.
doi: 10.1016/j.aml.2014.12.011. |
show all references
References:
[1] |
A. Apreutesei, A. Ducrot and V. Volpert,
Competition of species with intra-specific competition, Math. Model. Nat. Phenom., 3 (2008), 1-27.
doi: 10.1051/mmnp:2008068. |
[2] |
N. Apreutesei, A. Ducrot and V. Volpert,
Travelling waves for integro-differential equations in population dynamics, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), 541-561.
doi: 10.3934/dcdsb.2009.11.541. |
[3] |
N. Apreutesei, N. Bessonov, V. Volpert and V. Vougalter,
Sptaial structures and generalized travelling waves for an integro-differential equation, Discrete Contin. Dyn. Syst. Ser. B, 13 (2010), 537-557.
doi: 10.3934/dcdsb.2010.13.537. |
[4] |
O. Aydogmus,
Patterns and transitions to instability in an intraspecific competition model with nonlocal diffusion and interaction, Math. Model. Nat. Phenom., 10 (2015), 17-19.
doi: 10.1051/mmnp/201510603. |
[5] |
M. Banerjee and S. Banerjee,
Turing instabilities and spatio-temporal chaos in ratio-dependent Holling-Tanner model, Math. Biosci., 236 (2012), 64-76.
doi: 10.1016/j.mbs.2011.12.005. |
[6] |
M. Banerjee, N. Mukherjee and V. Volpert,
Prey-predator model with a nonlocal bistable dynamics of prey, Mathematics, 6 (2018), 1-13.
doi: 10.3390/math6030041. |
[7] |
M. Banerjee and S. Petrovskii, Self-organized spatial patterns and chaos in a ratio-dependent predator-prey system, Theor. Ecol., 4 (2011), 37-53. Google Scholar |
[8] |
M. Banerjee and V. Volpert, Prey-predator model with a nonlocal consumption of prey, Chaos, 26 (2016), 12pp.
doi: 10.1063/1.4961248. |
[9] |
M. Banerjee and V. Volpert, Spatio-temporal pattern formation in Rosenzweig-McArthur model: Effect of nonlocal interactions, Ecol. Complex., 30 (2017), 2-10. Google Scholar |
[10] |
M. Banerjee, V. Vougalter and V. Volpert,
Doubly nonlocal reaction–diffusion equations and the emergence of species, Appl. Math. Model., 42 (2017), 591-599.
doi: 10.1016/j.apm.2016.10.041. |
[11] |
M. Baurmann, W. Ebenhoh and U. Feudel,
Turing instabilities and pattern formation in a benthic nutrient-microorganism system, Math. Biosci. Eng., 1 (2004), 111-130.
doi: 10.3934/mbe.2004.1.111. |
[12] |
M. Baurmann, T. Gross and U. Feudel,
Instabilities in spatially extended predator-prey systems: Spatio-temporal patterns in the neighborhood of Turing-Hopf bifurcations, J. Theoret. Biol., 245 (2007), 220-229.
doi: 10.1016/j.jtbi.2006.09.036. |
[13] |
A. Bayliss and V. A. Volpert,
Patterns for competing populations with species specific nonlocal coupling, Math. Model. Nat. Phenom., 10 (2015), 30-47.
doi: 10.1051/mmnp/201510604. |
[14] |
N. Bessonov, N. Reinberg and V. Volpert,
Mathematics of Darwin's diagram, Math. Model. Nat. Phenom., 9 (2014), 5-25.
doi: 10.1051/mmnp/20149302. |
[15] |
S. Fasani and S. Rinaldi,
Factors promoting or inhibiting Turing instability in spatially extended prey-predator systems, Ecol. Model., 222 (2011), 3449-3452.
doi: 10.1016/j.ecolmodel.2011.07.002. |
[16] |
T. Galochkina, M. Marion and V. Volpert,
Initiation of reaction-diffusion waves of blood coagulation, Phys. D, 376-377 (2018), 160-170.
doi: 10.1016/j.physd.2017.11.006. |
[17] |
G. F. Gause, The Struggle for Existence, Williams and Wilkins, Baltimore, 1934.
doi: 10.5962/bhl.title.4489. |
[18] |
S. Genieys, N. Bessonov and V. Volpert,
Mathematical model of evolutionary branching, Math. Comput. Modelling, 49 (2009), 2109-2115.
doi: 10.1016/j.mcm.2008.07.018. |
[19] |
S. Genieys, V. Volpert and P. Auger,
Pattern and waves for a model in population dynamics with nonlocal consumption of resources, Math. Model. Nat. Phenom., 1 (2006), 65-82.
doi: 10.1051/mmnp:2006004. |
[20] |
S. Genieys, V. Volpert and P. Auger,
Adaptive dynamics: Modelling Darwin's divergence principle, Comp. Ren. Biol., 329 (2006), 876-879.
doi: 10.1016/j.crvi.2006.08.006. |
[21] |
J. D. Murray, Mathematical Biology. Ⅱ: Spatial Models And Biomedical Applications, Interdisciplinary Applied Mathematics, 19, Springer-Verlag, New York, 2003.
doi: 10.1007/b98869. |
[22] |
S. Pal, S. Ghorai and M. Banerjee,
Analysis of a prey-predator model with non-local interaction in the prey population, Bull. Math. Biol., 80 (2018), 906-925.
doi: 10.1007/s11538-018-0410-x. |
[23] |
S. V. Petrovskii and H. Malchow,
A minimal model of pattern formation in a prey-predator system, Math. Comput. Modelling, 29 (1999), 49-63.
doi: 10.1016/S0895-7177(99)00070-9. |
[24] |
J. A. Sherratt, B. T. Eagan and M. A. Lewis,
Oscillations and chaos behind predator-prey invasion: Mathematical artifact or ecological reality?, Phil. Trans. R. Soc. Lond. B, 352 (1997), 21-38.
doi: 10.1098/rstb.1997.0003. |
[25] |
V. Volpert, Branching and aggregation in self-reproducing systems, in MMCS, Mathematical Modelling of Complex Systems, ESAIM Proc. Surveys, 47, EDP Sci., Les Ulis, 2014,116–129.
doi: 10.1051/proc/201447007. |
[26] |
V. Volpert, Elliptic Partial Differential Equations, Monographs in Mathematics, 104, Birkhäuser/Springer Basel AG, Basel, 2014.
doi: 10.1007/978-3-0348-0813-2. |
[27] |
V. Volpert,
Pulses and waves for a bistable nonlocal reaction-diffusion equation, Appl. Math. Lett., 44 (2015), 21-25.
doi: 10.1016/j.aml.2014.12.011. |






[1] |
Alexandre Cornet. Mathematical modelling of cardiac pulse wave reflections due to arterial irregularities. Mathematical Biosciences & Engineering, 2018, 15 (5) : 1055-1076. doi: 10.3934/mbe.2018047 |
[2] |
Jian-Jhong Lin, Weiming Wang, Caidi Zhao, Ting-Hui Yang. Global dynamics and traveling wave solutions of two predators-one prey models. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1135-1154. doi: 10.3934/dcdsb.2015.20.1135 |
[3] |
Johannes Lankeit, Yulan Wang. Global existence, boundedness and stabilization in a high-dimensional chemotaxis system with consumption. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6099-6121. doi: 10.3934/dcds.2017262 |
[4] |
Guo Lin, Wan-Tong Li. Traveling wave solutions of a competitive recursion. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 173-189. doi: 10.3934/dcdsb.2012.17.173 |
[5] |
Johan Henriksson, Torbjörn Lundh, Bernt Wennberg. A model of sympatric speciation through reinforcement. Kinetic & Related Models, 2010, 3 (1) : 143-163. doi: 10.3934/krm.2010.3.143 |
[6] |
Andrei Korobeinikov. Global properties of a general predator-prey model with non-symmetric attack and consumption rate. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 1095-1103. doi: 10.3934/dcdsb.2010.14.1095 |
[7] |
Jiashan Zheng, Yifu Wang. A note on global existence to a higher-dimensional quasilinear chemotaxis system with consumption of chemoattractant. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 669-686. doi: 10.3934/dcdsb.2017032 |
[8] |
Jian Yang, Bendong Lou. Traveling wave solutions of competitive models with free boundaries. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 817-826. doi: 10.3934/dcdsb.2014.19.817 |
[9] |
Fathi Dkhil, Angela Stevens. Traveling wave speeds in rapidly oscillating media. Discrete & Continuous Dynamical Systems - A, 2009, 25 (1) : 89-108. doi: 10.3934/dcds.2009.25.89 |
[10] |
Bingtuan Li. Some remarks on traveling wave solutions in competition models. Discrete & Continuous Dynamical Systems - B, 2009, 12 (2) : 389-399. doi: 10.3934/dcdsb.2009.12.389 |
[11] |
Wei Ding, Wenzhang Huang, Siroj Kansakar. Traveling wave solutions for a diffusive sis epidemic model. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1291-1304. doi: 10.3934/dcdsb.2013.18.1291 |
[12] |
Vishal Vasan, Katie Oliveras. Pressure beneath a traveling wave with constant vorticity. Discrete & Continuous Dynamical Systems - A, 2014, 34 (8) : 3219-3239. doi: 10.3934/dcds.2014.34.3219 |
[13] |
Jong-Shenq Guo, Ken-Ichi Nakamura, Toshiko Ogiwara, Chang-Hong Wu. The sign of traveling wave speed in bistable dynamics. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 0-0. doi: 10.3934/dcds.2020047 |
[14] |
Hongyong Zhao, Daiyong Wu. Point to point traveling wave and periodic traveling wave induced by Hopf bifurcation for a diffusive predator-prey system. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020129 |
[15] |
Zuo Quan Xu, Fahuai Yi. An optimal consumption-investment model with constraint on consumption. Mathematical Control & Related Fields, 2016, 6 (3) : 517-534. doi: 10.3934/mcrf.2016014 |
[16] |
Wei-Jian Bo, Guo Lin, Shigui Ruan. Traveling wave solutions for time periodic reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4329-4351. doi: 10.3934/dcds.2018189 |
[17] |
M. B. A. Mansour. Computation of traveling wave fronts for a nonlinear diffusion-advection model. Mathematical Biosciences & Engineering, 2009, 6 (1) : 83-91. doi: 10.3934/mbe.2009.6.83 |
[18] |
Zhaosheng Feng, Goong Chen. Traveling wave solutions in parametric forms for a diffusion model with a nonlinear rate of growth. Discrete & Continuous Dynamical Systems - A, 2009, 24 (3) : 763-780. doi: 10.3934/dcds.2009.24.763 |
[19] |
Junhao Wen, Peixuan Weng. Traveling wave solutions in a diffusive producer-scrounger model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 627-645. doi: 10.3934/dcdsb.2017030 |
[20] |
Cunming Liu, Jianli Liu. Stability of traveling wave solutions to Cauchy problem of diagnolizable quasilinear hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4735-4749. doi: 10.3934/dcds.2014.34.4735 |
2018 Impact Factor: 0.545
Tools
Metrics
Other articles
by authors
[Back to Top]